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We propose aC0interior penalty method for the fourth-order stream function formulation of the surface Stokes problem. The scheme utilizes continuous, piecewise polynomial spaces defined on an approximate surface. We show that the resulting discretization is positive definite and derive error estimates in various norms in terms of the polynomial degree of the finite element space as well as the polynomial degree to define the geometry approximation. A notable feature of the scheme is that it does not explicitly depend on the Gauss curvature of the surface. This is achievedviaa novel integration-by-parts formula for the surface biharmonic operator.
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Inversion formula and range conditions for a linear system related with the multi‐interval finite Hilbert transform in L 2
Abstract Givenndisjoint intervals on together withnfunctions , , and an matrix , the problem is to find anL2solution , , to the linear system , where , is a matrix of finite Hilbert transforms with defined on , and is a matrix of the corresponding characteristic functions on . Since we can interpret , as a generalized multi‐interval finite Hilbert transform, we call the formula for the solution as “the inversion formula” and the necessary and sufficient conditions for the existence of a solution as the “range conditions”. In this paper we derive the explicit inversion formula and the range conditions in two specific cases: a) the matrix Θ is symmetric and positive definite, and; b) all the entries of Θ are equal to one. We also prove the uniqueness of solution, that is, that our transform is injective. In the case a), that is, when the matrix Θ is positive definite, the inversion formula is given in terms of the solution of the associated matrix Riemann–Hilbert Problem. In the case b) we reduce the multi interval problem to a problem onncopies of and then express our answers in terms of the Fourier transform. We also discuss other cases of the matrix Θ.
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- PAR ID:
- 10450121
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Mathematische Nachrichten
- Volume:
- 294
- Issue:
- 8
- ISSN:
- 0025-584X
- Page Range / eLocation ID:
- p. 1523-1546
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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